152 XIII INTEGRATION definition of a martingale; now let q-* -f oo, thenp-^- + oo, and so obtain a contra- diction.) (b) Deduce from (a) that the sequence (/„) converges almost everywhere to an integrable function/. (Give a, b rational values, and use Fatou's lemma.) (c) Give an example where \fd^ is not equal to the (constant) value of the integrals !/„ dp,. (Take X = [0, 1] and take ^ to be Lebesgue measure; take wn to consist of the intervals [0, 2~nl [2~n, 2-"+1[, . . . , [J, 1].) (d) Let F(x) = supfn(x). Show that there exists a constant C > 0 such that, for all n a > 0, if B*7 is the set of points x such that F(x) > a, we have ^t(B«) ^ C/a. (Remark that Ba is the union of the sets Ba, „ = {x e X:fn(x) > a}.) 26. With the hypotheses of Problem 25, let a be another finite partition of X into /z- integrable sets. For each x e X and each integer n, let B 6 or and A e wn be the sets of the partitions a and wn which contain x ; put fn(x) = 0 if ft(A) = 0, and fn(x) = /x(B n A)/^u(A) if /z(A) > 0. Show that the sequence (/„(*)) converges almost every- where to a limit /(x) <[ 1, and that the sequence (/„) converges in mean to/. (For each set B e a, observe that the functions cpBfn form an elementary martingale for the measure <pB - ji/,, and use Problem 25.) 27. Let X be a compact space with measure fi(X) = 1, and let a = (AOi^^n be a finite partition of X into integrable subsets. For each x e X, if At e a is the set containing x, the number j(a;x)= -logft(A,) is called the information at the point jc corresponding to the partition a. The number H(a) = f/(«; J ^ 0 (with the convention that t log t = 0 when f = 0) is called the entropy of the partition a (relative to the measure JJL). It is the " average" of the information corresponding to a. If ^8 = (Bk)i^k^m is another finite partition of X into integrable subsets, the number H(a/]8) = - S M(Ai n B,) log(^(A, n Bfc)//z(B*)) ^ 0 (in which the terms for which ft(Bfc) = 0 are replaced by 0) is called the entropy of a relative to ft. If a> is the partition consisting of the single set X, then H(a/co) = H(a). (a) If (otj)i^j<n is a finite sequence of finite partitions of X into integrable sets, we denote by V aj or #1 V oc2 V . . . V a« the coarsest of the partitions which are finer j=i than all the a/ , that is to say the partition consisting of all nonempty intersections A! n A2 n • • • n A« , where Aj e a,- for each j. If a, /?, y are three finite partitions of X into integrable sets, show that H((a V j0)/y) - H(a/y) -f H(£/(a V and in particular H(a V j8) = H(a).+ H(j8/a).ach set A e w» the value of/„ is equal to the average of/m over A),